Optimal. Leaf size=68 \[ \frac{219 x+89}{276 \left (2 x^2-x+3\right )^{3/2}}-\frac{2604 x+1465}{2116 \sqrt{2 x^2-x+3}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{2}} \]
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Rubi [A] time = 0.0518717, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {1660, 12, 619, 215} \[ \frac{219 x+89}{276 \left (2 x^2-x+3\right )^{3/2}}-\frac{2604 x+1465}{2116 \sqrt{2 x^2-x+3}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{2+x+3 x^2-x^3+5 x^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=\frac{89+219 x}{276 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{-\frac{159}{16}+\frac{207 x}{8}+\frac{345 x^2}{4}}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac{89+219 x}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{1465+2604 x}{2116 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{7935}{16 \sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=\frac{89+219 x}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{1465+2604 x}{2116 \sqrt{3-x+2 x^2}}+\frac{5}{4} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{89+219 x}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{1465+2604 x}{2116 \sqrt{3-x+2 x^2}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{4 \sqrt{46}}\\ &=\frac{89+219 x}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{1465+2604 x}{2116 \sqrt{3-x+2 x^2}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.226112, size = 55, normalized size = 0.81 \[ \frac{5 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{4 \sqrt{2}}-\frac{7812 x^3+489 x^2+7002 x+5569}{3174 \left (2 x^2-x+3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 146, normalized size = 2.2 \begin{align*} -{\frac{5\,{x}^{3}}{6} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{{x}^{2}}{8} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{47\,x}{64} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{271}{768} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{-2423+9692\,x}{17664} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{-173+692\,x}{1587}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{5\,x}{4}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{5}{16}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{5\,\sqrt{2}}{8}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.46495, size = 250, normalized size = 3.68 \begin{align*} \frac{5}{6348} \, x{\left (\frac{284 \, x}{\sqrt{2 \, x^{2} - x + 3}} - \frac{3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{71}{\sqrt{2 \, x^{2} - x + 3}} + \frac{805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\right )} + \frac{5}{8} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{355}{3174} \, \sqrt{2 \, x^{2} - x + 3} - \frac{58 \, x}{1587 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{x^{2}}{2 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1897}{6348 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{95 \, x}{276 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{41}{276 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.34762, size = 300, normalized size = 4.41 \begin{align*} \frac{7935 \, \sqrt{2}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) - 8 \,{\left (7812 \, x^{3} + 489 \, x^{2} + 7002 \, x + 5569\right )} \sqrt{2 \, x^{2} - x + 3}}{25392 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x^{2} - x + 3\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14788, size = 84, normalized size = 1.24 \begin{align*} -\frac{5}{8} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac{3 \,{\left ({\left (2604 \, x + 163\right )} x + 2334\right )} x + 5569}{3174 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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